Proof
By repeated mean-proportional construction (VI.13) on the medial,
each new line is irrational with respect to all earlier ones (using
the unique-decomposition results X.42–X.47, X.79–X.84).
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Depends on (3)
- VI.13Proposition VI.13To two given straight lines to find a mean proportional.
- X.21Proposition X.21The rectangle contained by rational straight lines commensurable in square only is irrational, and the side of the…
- X.114Proposition X.114If an area be contained by an apotome and the binomial the terms of which are commensurable with the terms of the…
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